We have developed vertical seismic wave gradiometry (VSWG) to estimate velocity, impedance, and attenuation structure in the vicinity of boreholes using borehole array waveforms of check-shot explosions near the borehole head. We have extended wave gradiometry (WG) theory from a purely local relation of the wavefield and wavefield spatial gradient to one that incorporated the ray ansatz over the length of the borehole for a traveling wave in a vertically inhomogeneous medium. We checked the ray assumption against acoustic full-wave synthetic seismograms, and it was found to yield robust measures of the medium velocity. Anelastic attenuation and impedance structure trade off, but in cases of high anelastic attenuation, realistic bounds can be placed on the seismic impedance that effectively constrains the average attenuation. We have applied these methods to data collected at the San Andreas Fault Observatory at Depth borehole in central California in 2005, and we found that the WG velocity estimate agreed well with the borehole acoustic log and previously determined vertical seismic profile results based on traveltime analysis except at the bottom of the hole, where refraction on a near-vertical fault affected the data. The average is approximately 20 in the frequency band of 10–30 Hz, and it is required to yield realistic values of impedance in the local medium. Application of VSWG yields appropriate, smoothed velocity models that can be used as an end product or as a starting model for full-wave inversion.